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Math Help - Smallest Positive integer n

  1. #1
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    Smallest Positive integer n

    Find the form of all n in N satifying tow(n)=6. (sorry don't know how to write this Tex). What is the smallest positive integer n for which this is true?

    tow(n)=6 so 6=(1+a1)(1+a2)----(1+ak) where n=p1^a1p2^a2---pk^ak
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  2. #2
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    \tau(n)=6
    We have \tau(n)=(1+\alpha_{1})(1+\alpha_{2}).....(1+\alpha  _{k})for k= p^{\alpha_{1}}p^{\alpha_{2}}...p^{\alpha_{k}}
    So 6=(1+\alpha_{1})(1+\alpha_{2}).....(1+\alpha_{k})
    I just don't know about the next step...
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  3. #3
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    I tried different values of n:
    Well 2^5 works
    (5+1)=6
    3^1*7^2
    (1+1)(2+1)=2*3=6

    So n=p^5 or p1^2*p2?
    And 12 is the smallest n?
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  4. #4
    Senior Member Shanks's Avatar
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    hint: since 6 has only one non-trival factorization 6=2*3, we obtain k=2, and a_1 = 1, a_2 = 2.
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